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Cambridge University Press 978-0-521-11842-2 - Lectures on Algebraic Cycles, Second Edition Spencer Bloch Frontmatter More information

Lectures on Algebraic Cycles Second Edition

Spencer Bloch’s 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch–Kato conjecture on special values of zeta functions. The book begins with Mumford’s example showing that the Chow group of zero-cycles on an algebraic variety can be infinite dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena.

spencer bloch is R. M. Hutchins Distinguished Service Professor in the Department of Mathematics at the University of Chicago.

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1 M. Cabanes and M. Enguehard Representation Theory of Finite Reductive Groups 2 J. B. Garnett and D. E. Marshall Harmonic Measure 3 P. Cohn Free Ideal Rings and Localization in General Rings 4 E. Bombieri and W. Gubler Heights in Diophantine Geometry 5 Y. J. Ionin and M. S. Shrikhande Combinatorics of Symmetric Designs 6 S. Berhanu, P. D. Cordaro and J. Hounie An Introduction to Involutive Structures 7 A. Shlapentokh Hilbert’s Tenth Problem 8 G. Michler Theory of Finite Simple Groups I 9 A. Baker and G. Wustholz¨ Logarithmic Forms and Diophantine Geometry 10 P. Kronheimer and T. Mrowka Monopoles and Three-Manifolds 11 B. Bekka, P. de la Harpe and A. Valette Kazhdan’s Property (T) 12 J. Neisendorfer Algebraic Methods in Unstable Homotopy Theory 13 M. Grandis Directed Algebraic Topology 14 G. Michler Theory of Finite Simple Groups II 15 R. Schertz Complex Multiplication

Lectures on Algebraic Cycles Second Edition

SPENCER BLOCH University of Chicago

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao˜ Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York

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C S. Bloch 2010 C Mathematics Department, Duke University 1980

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Contents

Preface to the second edition page vii 0 Introduction 3 1 Zero-cycles on surfaces 9 Appendix: On an argument of Mumford in the theory of algebraic cycles 21 2 Curves on threefolds and intermediate jacobians 25 3 Curves on threefolds – the relative case 37 4 K-theoretic and cohomological methods 45 5 Torsion in the Chow group 59 2 6 Complements on H (K2) 69 7 Diophantine questions 79 8 Relative cycles and zeta functions 97 9 Relative cycles and zeta functions – continued 115 Bibliography 123 Index 129

Preface to the second edition

30 Years later...

Looking back over these lectures, given at Duke University in 1979, I can say with some pride that they contain early hints of a number of important themes in modern arithmetic geometry. Of course, the flip side of that coin is that they are now, thirty years later, seriously out of date. To bring them up to date would involve writing several more monographs, a task best left to mathematicians thirty years younger than me. What I propose instead is to comment fairly briefly on several of the lectures in an attempt to put the reader in touch with what I believe are the most important modern ideas in these areas. The section on motives just below is intended as a brief introduction to the modern viewpoint on that subject. The remaining sections until the last follow roughly the content of the original book, though the titles have changed slightly to reflect my current emphasis. The last section, motives in physics, represents my recent research.

In the original volume I included a quote from Charlie Chan, the great Chi- nese detective, who told his bumbling number one son “answer simple, but question very very hard.” It seemed to me an appropriate comment on the subject of algebraic cycles. Given the amazing deep new ideas introduced into the subject in recent years, however, I think now that the question remains very very hard, but the answer is perhaps no longer so simple...

At the end of this essay I include a brief bibliography, which is by no means complete. It is only intended to illustrate the various ideas mentioned in the text.

viii Preface to the second edition Motives

Much of the recent work in this area is centered around motives and the construction – in fact various constructions, due to Hanamura (1995), Levine (1998), and Voevodsky (Mazza et al. 2006; Voevodsky et al. 2000) – of a triangulated category of mixed motives. I will sketch Voevodsky’s construction as it also plays a central role in his proof of the Bloch–Kato–Milnor conjecture discussed in Lecture 5. Then I will discuss various lectures from the original monograph. Let k be a field. The category Cork is an additive category with objects = → smooth k-varieties: Ob(Cork) Ob(Smk). Morphisms Z : X Y are finite linear combinations of correspondences Z = niZi where Zi ⊂ X × Y is closed and the projection πi : Zi → X is finite and surjective. Intuitively, we may think → n ∈ −1 of Zi as a map X Sym Y associating to x X the fibre fi (x)viewedasa zero-cycle on Y. There is an evident functor

Smk → Cork which is the identity on objects. A presheaf on a category C with values in an abelian category A is simply a op contravariant functor F : C →A.AnA-valued presheaf F on Cork induces | a presheaf F Smk on Smk. Intuitively, to lift a presheaf G from Smk to Cork one needs a structure of trace maps or transfers f∗ : G(Z) → G(X)forZ/X ﬁnite. Presheaves on Cork are referred to as presheaves with transfers. For X ∈ Ob(Smk) one has the representable sheaf Ztr(X) deﬁned by = , . Ztr(X)(U) HomCork (U X)

An important elaboration on this idea yields for pointed objects xi ∈ Xi

Ztr((X1, x1) ∧···∧(Xn, xi)) := Coker Ztr(X1 ×···Xi ×···Xn) → Ztr( Xi) . n 1 In particular, one deﬁnes Ztr( Gm) by taking Xi = A −{0} and xi = 1. A presheaf with transfers F is called homotopy invariant if, with obvious ∗ = ∗ × 1 → notation, i0 i1 : F(U A ) F(U). The complex of chains C∗(F)ona presheaf with transfers F is the presheaf of complexes (placed in cohomological degrees [−∞, 0])

n 0 C∗(F):= U →···→ F(U × ∆ ) →···→ F(U × ∆ ) Here n (0.1) ∆ := Spec k[t0,...,tn]/( ti − 1)

Preface to the second edition ix

is the algebro-geometric n-simplex. The boundary maps in the complex are the usual alternating sums of restrictions to the faces ∆n−1 → ∆n deﬁned by setting ti = 0. The two restrictions ∗, ∗ × 1 → i0 i1 : C∗(F)(U A ) C∗(F)(U)

are shown to be homotopic, so the homology presheaves Hn(C∗(F)) are homotopy invariant. 1 1 Maps f0, f1 : X → Y in Cork are A -homotopic if there exists H : X×A → Y = ∗ 1 1 in Cork such that f j i j H. A -homotopy is an equivalence relation, and A - homotopic maps induce homotopic maps

f0∗ f1∗ : C∗Ztr(X) → C∗Ztr(Y). Voevodsky deﬁnes q Z(q):= C∗Ztr( Gm)[−q], q ≥ 0. More precisely, the above complex is viewed as a complex of presheaves on Smk and then localized for the Zariski topology. Motivic cohomology is then deﬁned (for q ≥ 0) as the hypercohomology of this complex of Zariski sheaves: p , = p , . HM(X Z(q)) : HZar(X Z(q))

One has a notion of tensor product for presheaves on the category Cork, and Ztr(X) ⊗ Ztr(Y ) = Ztr(X × Y ). In particular, Z(p) ⊗ Z(q) → Z(p + q) so one gets a product structure on motivic cohomology. In low degrees one has 0 , = π , HM(X Z(0)) Z[ 0(X)] p , = , > , HM(X Z(0)) (0) p 0

Z(1) Gm[−1]. Another important sign that this is the right theory is the link with Milnor Milnor × K-theory. (K∗ (k) is defined as the quotient of the tensor algebra on k by the ideal generated by quadratic relations a ⊗ (1 − a)fora ∈ k −{0, 1}.) n , Milnor Theorem HM(Spec k Z(n)) Kn (k). The fact that the Zariski topology suffices to define motivic cohomology is somewhat surprising because a Zariski open cover π: U → X does not yield a resolution of Zariski sheaves

(0.2) ···→Ztr(U ×X U) → Ztr(U) → Z(X) → 0. To remedy this, Voevodsky employs the Nisnevich topology. A morphism π: U → X is a Nisnevich cover if for any ﬁeld K/k one has U(K) X(K).

x Preface to the second edition

To see that (0.2) becomes exact when localized for the Nisnevich topology, one uses the fact that any finite cover of a Hensel local ring is a product of local rings. The actual triangulated category of effective motives over k is a quotient cat- − egory of the derived category D (ShNis(Cork)) of bounded-below complexes of Nisnevich sheaves on Cork. One considers the smallest thick subcategory W 1 containing all cones of Ztr(X × A ) → Ztr(X), and one defines eff = − −1 . DMNis(k): D ShNis(Cork)[W ] Said another way, one formally inverts all morphisms with cones in W. Finally, the motive associated to a smooth k-variety X is defined by = ∈ eff . (0.3) M(X): Ztr(X) DMNis(k)

eﬀ eﬀ The category of geometric motives DMgeo(k) is the thick subcategory in DMNis(k) generated by the M(X).

One has the following properties:

Mayer-Vietoris

M(U ∩ V) → M(U) ⊕ M(V) → M(X) → M(U ∩ V)[1]

is a distinguished triangle.

Kunneth¨ M(X × Y ) = M(X) ⊗ M(Y ).

Vector bundle theorem M(X) M(V)

for V/X a vector bundle.

Cancellation Assume varieties over k admit a resolution of singularities. Write M(q):= M ⊗ Z(q). Then

Hom(M, N) Hom(M(q), N(q)).

The category of (not necessarily eﬀective) motives is obtained by inverting the functor M → M(1) in DMeﬀ.

Projective bundle theorem For V/X a rank n + 1 vector bundle n . M(P(V)) i=0 M(X)(i)[2i]

Preface to the second edition xi

Chow motives If X, Y are smooth projective, then

(0.4) Hom(M(X), M(Y )) CHdimX(X × Y ).

The category of Chow motives over a ﬁeld k has as objects triples dimX (X, p, m) with X smooth projective over k, p ∈ CH (X × X)Q a projector (i.e. p ◦ p = p) and m ∈ Z. The morphisms are given by

(0.5) Hom((X, p, m), (Y, q, n)) := q ◦ CHdimX+n−m(X × Y ) ◦ p.

eﬀ It follows from (0.4) and the existence of projectors in DMNis(k) that the cate- eﬀ gory of Chow motives embeds in DMNis(k). Motivic cohomology For X/k smooth, we have

p (0.6) H (X, Z(q)) Hom eff (Ztr(X), Z(q)[p]). M DMNis(k) In fact, motivic cohomology is closely related to algebraic cycles, and this relationship lies at the heart of modern cycle theory. There are a number of ways to formulate things. I will use higher Chow groups because they relate most naturally to arithmetic questions. Let ∆• be the cosimplicial variety as in (0.1) above. Define Zq(X, n) to be the free abelian group of algebraic cycles on X × ∆n which are in good position with respect to all faces X × ∆m → X × ∆n. The complex Zq(X, •) is defined by taking alternating sums of pullbacks in the usual way:

(0.7) ···→Zq(X, 2) →Zq(X, 1) →Zq(X, 0) → 0.

(Here Zq(X, n) is placed in cohomological degree −n.) The higher Chow groups are deﬁned by

(0.8) CHq(X, n):= H−n(Zq(X, •)).

For example, the usual Chow group CHq(X) = CHq(X, 0). Voevodsky proves that for X smooth over a perfect ﬁeld k one has

p , q , − = p Zq , • − . (0.9) HM(X Z(q)) CH (X 2q p) H ( (X )[ 2q]) Beilinson and Soule´ conjecture that the shifted chain complex Zq(X, •)[−2q] ⊗ Q has cohomological support in degrees [0, 2q]. Actually, their conjecture was formulated in terms of the γ-ﬁltration in K-theory, but one has the further identiﬁcation

p , ⊗ q , − ⊗ q ⊗ . (0.10) HM(X Z(q)) Q CH (X 2q p) Q grγKp(X) Q

xii Preface to the second edition Lecture 1: Zero-cycles

The two most important ideas here are ﬁrstly the conjecture that surfaces with geometric genus zero (pg = 0) should have Chow group of zero-cycles representable. For S such a surface over C we expect an exact sequence

deg 0 → Alb(S ) → CH0(S ) −−→ Z → 0.

The group T(S ) defined in Lemma 1.4 is conjectured to be zero in this case. Secondly, for any smooth projective variety X,theChow group of zero-cycles CH0(X) is conjectured (1.8) to carry a descending ∗ filtration F CH0(X) which is functorial for correspondences such that Λ p −→ p the map grF CH0(X) grF CH0(Y ) induced by an algebraic cycle Λ ∈ CHdimY (X × Y ) depends only on the class of Λ in cohomology. Indeed, one may conjecture the existence of such a filtration on CHq(X) for any q. These conjectures remain unproven, but a very beautiful general picture, based on the yoga of mixed motives, has been elaborated by A. Beilinson. Interested readers should consult the important article by Jannsen (1994) and the literature cited there. Let me sketch briefly (following Jannsen) the modern viewpoint. It is convenient to dualize the definition of M(X) (0.3). Assume X smooth, projective of dimension d. Define (Hom means the internal Hom in DM) ∗ = , . M(X) HomDM(M(X) Z(0)) The formula for the Chow group becomes p = 2p , = , ∗ . CH (X) HM (X Z(p)) HomDM(Z(0) M(X) (p)[2p]) One of Grothendieck’s standard conjectures about algebraic cycles is that there exist Künneth projectors

d πi ∈ CH (X × X)Q/{homological equivalence}

inducing the natural projections H∗(X) → Hi(X) on cohomology. If we assume d further that the ideal {homological equiv.}/{rational equiv.}⊂CH (X × X)Q is

nilpotent (nilpotence conjecture), then the pri lift (non-canonically) to projec- ∈ d × ∗ ⊗ = tors pri,rat CH (X X)Q and we may use (0.5) to decompose M(X) Q i − i h [ i] non-canonically as a direct sum of Chow motives. This idea is due to J. Murre (1993a). The hope is that DM admits a t-structure such that

(0.11) Hi(M(X)∗ ⊗ Q) = hi(X).

Preface to the second edition xiii

The resulting spectral sequence

p,q = , q ∗ ⇒ , ∗ + E2 HomDM(Z(0) H (M(X) ( j)[p]) HomDM(Z(0) M (X)( j)[p q]) would yield ﬁltrations on the Chow groups ⊗Q with ν j 2 j−ν 2 j−i , i , F CH (X)Q i=0 ExtDM (Q(0) h (X)( j)) ν j ν , 2 j−ν . grF CH (X)Q ExtDM(Q(0) H (X)( j)) Murre suggests a natural strengthening of his conjectures, based on the idea that one should be able for i ≤ d, to ﬁnd representatives for πi supported on Xi × X ⊂ X × X, where Xi ⊂ X is a general plane section of dimension i.For j example, π0 = {x}×X for a point x. Clearly this would imply πiCH (X) = 0 for i < j, and since the conjectures imply ν j = 2 j−ν π j , F CH (X) i=0 i CH (X) we could conclude further that FνCH j(X) = (0),ν>j.

Suppose, for example, that dim X = 2. We would get a 3-step filtration on 0 1 2 the zero-cycles: CH0(X) = F ⊃ F ⊃ F ⊃ (0), with 0 = , 4 , grF CH0(X)Q HomDM(Q(0) h (X)(2)) 1 = 1 , 3 , grF ExtDM(Q(0) h (X)(2)) 2 = 2 , 2 . grF ExtDM(Q(0) h (X)(2)) This fits perfectly with the ideas in Lecture 1. Indeed, Murre has computed gr0 and gr1 and he finds exactly the degree and the Albanese. Of course, gr2 is more problematical, but note that the condition discussed in the text that 2 H (X, Q(1)) should be generated by divisors (which is equivalent to pg = 0 in characteristic zero) means h2(X)(2) ⊕Q(1). (This is obvious for motives modulo homological equivalence. Assuming the nilpotence conjecture, it holds 2 = 2 for Chow motives as well.) In this case, grF can be computed for X P when it is clearly (0). The conjectural “theorem of the hypersquare” (Proposition 1.12) can be understood in motivic terms (Jannsen 1994, conj. 3.12) using the fact that n n h (X0 ×···×Xn) is a direct summand of h (X0 ×···Xi ···×Xn). Clearly wrongheaded, however, is Metaconjecture 1.10, which stated that 2 2 F CH0(X) is controlled by the polarized Hodge structure associated to H (X). Indeed, Ext2 = (0) in the category of Hodge structures. One may try (compare Carlson and Hain 1989) to look at Exts in some category of variations of Hodge

xiv Preface to the second edition

structure. In the absence of parameters supporting such variations (i.e. for X over a number ﬁeld), however, the Ext2 term should vanish and we should have F2CH2(X) = (0).

Lectures 2 and 3: Intermediate jacobians

The modern point of view about intermediate jacobians is to view them as Ext1(Z(0), H) where H is a suitable Hodge structure, and the Ext group is taken in the abelian category of mixed Hodge structures (Carlson 1987). In the classic situation, H = H2r−1(X, Z(r)) where X is a smooth projective variety. Note in this case that H has weight −1. An extension 0 → H → E → Z(0) → 0 would yield a mixed Hodge structure E with weights 0, −1 and Hodge ﬁltration

−1 0 EC = F EC ⊃ F EC ··· .

Let f, w ∈ EC be liftings of 1 ∈ Z(0) splitting the weight and Hodge filtrations respectively. The difference between them w − f gives a well-defined class in = / 0 + J HC (F HC HZ) which is the intermediate jacobian. To define the class of a codimension-r cycle Z = ni Zi,let|Z| be the support of the cycle. We → 2r , have a cycle class with supports [Z]: Z H|Z|(X Z(r)) and a diagram

(0.12) 2r−1 , 2r−1 , 2r−1 −| |, H|Z| (X Z(r)) H (X Z(r)) H (X Z Z(r))

2r , 2r , H|Z|(X Z(r)) H (X Z(r)) |Z| Z 0 H2r(X, Z(r))

2r−1 , The ﬁrst group H|Z| (X Z(r)) is zero by purity, and vanishing of the lower right-hand arrow will hold if Z is homologous to 0. Assuming this, we get the desired extension of Hodge structures. (I believe this construction is due to Deligne, though I do not have a good reference.) For H any mixed Hodge structure with weights < 0 one has an analogous construction. Note, however, the resulting abelian Lie group J need not be compact. For example, Ext1(Z(0), Z(1)) C× S 1 × R. In Lecture 3, the focus is on the case

= 1 , ⊗ 1 , ⊗2 ⊂ 3 × 2, . H H (C Z(2)) Hc (Gm Z) Hc (C (Gm) Z(2))

Preface to the second edition xv

Here H has weight −3, and 1 , 1 , / 1 , 1 , × . ExtMHS (Z(0) H) H (C C) H (C Z(2)) H (C C (1)) At the time I was very much inspired by the work of Borel (1977) on regulators for higher K-groups of number fields. I believed that similar regulators could be defined for arithmetic algebraic varieties more generally, and that these regulators could be related to values of Hasse–Weil L-functions. This was done in a very limited and ad hoc way in Bloch (1980, 2000), and then much more definitively by Beilinson (1985). From the point of view of Lecture 3, the regulator can be thought of as a relative cycle class map

p , (0.9) q , − → 1 , = / + 0 . HM(X Z(q)) CH (X 2q p) ExtMHS (Z(0) H) HC (HZ F HC) Here H = H2q−1(X × ∆2q−p, X × ∂∆2q−p; Z(q)). For details of this construction, see Bloch (2000). Other constructions are given in Bloch (1986b) and Goncharov (2005). The quotient of this Ext group by its maximal compact subgroup is the cor- responding Ext in the category of R-Hodge structures. It is an R-vector space. More generally one can associate to any mixed Hodge structure a nilpotent matrix γ (Cattani et al., 1986, prop. 2.20), which is the obstruction to a real splitting of the ﬁltration by weights. These invariants arise, for example, if the curve C in Lecture 3 is allowed to degenerate, so H1(C, Z) is itself a mixed Hodge structure.

Lecture 4: Cohomological methods

This chapter contains basic information about algebraic K-theory, an important tool in the study of algebraic cycles. I describe the “Quillen trick” and use it to construct the Gersten resolution for K-sheaves and also Betti and etale´ cohomology sheaves for smooth varieties. Brieﬂy, one considers Zariski sheaves K H q H q → q (resp. Betti, resp. et) associated to the presheaf U Kq(U) of algebraic → q , → q , / K-groups (resp. U HBetti(U Z), resp. U Het(U Z nZ)). One obtains resolutions of these sheaves which enable one to identify, for example,

p p H (X, Kp) CH (X), (0.13) p , H p p /{ }. H (X Betti) CH (X) algebraic equivalence I think it is fair to say that the resulting K-cohomology and the parallel constructions for Betti and etale´ cohomology have had important technical applications but have not been the breakthrough one had hoped for at the time.

xvi Preface to the second edition

Despite the Gersten resolution, it turns out to be difficult to interpret the resulting cohomology. Finiteness results, for example, are totally lacking. One nice application of the Betti theory (see reference [6] at the end of Lecture 4) was to falsify a longstanding conjecture about differential forms of the second kind ≥ p,q = p , H q → on varieties of dimension 3. The spectral sequence E2 HZar(X Betti) p+q , HBetti(X Z) leads to an exact sequence 3 −→a Γ , H 3 −→b 2 , H 2 −→c 4 (0.14) HBetti(X) (X ) H (X ) HBetti(X) Using (0.13) one can identify Ker(c) in (0.14) with the Griffiths group of cycles homologous to zero modulo algebraic equivalence, a group which in some cases is known not to be finitely generated (Clemens 1983). It follows in such cases that a is not surjective, indeed Coker(a) is infinitely generated. But Γ(X, H 3) is precisely the space of meromorphic 3-forms of the second kind; that is meromorphic forms which at every point differ from an algebraic form which is regular at the point by an exact algebraic form. Thus, unlike the case of curves and surfaces, differential forms of the second kind do not necessarily come from global cohomology classes in dimensions ≥ 3.

Lecture 5: The conjecture of Milnor–Bloch–Kato

M Let F be a ﬁeld and a prime with 1/ ∈ K. The Milnor ring K∗ (F)/ is generated by F×/F× with relations given by Steinberg symbols f ⊗(1− f ), f 0, 1. The conjecture in question states that the natural map to Galois cohomology

M ∗ ∗ K∗ (F)/ → H (F,µ) is an isomorphism. My own contribution to this, which is explained in Lec- M → n ,µ⊗n ture 5, is a proof that Kn (F) H (F ) is surjective when F has cohomological dimension n. For some years Voevodsky has been working on a very difficult program, using his own motivic theory and results of M. Rost, to prove the conjecture in complete generality. The proof is now complete (for an outline with references, see the webpage of C. Weibel), but there is still no unified treatment and the arguments use sophisticated techniques in algebraic homotopy theory which I have not understood. Geometrically, the result can be formulated as follows. Let i: X → Y be a closed immersion of varieties over a field k of characteristic prime to , and let j: Y − X → Y be the open immersion. For simplicity I assume µ ⊂ k so there is no need to distinguish powers of µ. Consider the exact sequence ∗ ∂ p i p p+1 H (Y, Z/) −→ H (X, Z/) −→ H (Y, j!Z/).

Preface to the second edition xvii

Given a cohomology class c ∈ H p(X) and a smooth point x ∈ X, there exists a ⊃ = ∂ ∈ p+1 , / Zariski open neighborhood Y U x such that 0 (c) U H (U j!Z ). As an exercise, the reader might work out how this is equivalent to the conjecture for F = k(X). Another exercise is to formulate a version of coniveau p+1 filtration as described on page 52 for the group H (Y, j!Z/) in such a way that the image of ∂ lies in F1. The whole picture of motivic cohomology with finite coefficients is now quite beautiful (Voevodsky et al. 2000). Let X be a smooth, quasi-projective scheme over an algebraically closed field k, and let m ≥ 2 be relatively prime to the characteristic. Let r ≥ dim X. Then 2r−n , / 2r−n , / . HM (X Z mZ(r)) Het (X Z mZ(r)) Said another way, the cycle complexes Zr(X, •)[−2r], in equation (0.9), com- pute the -adic etale´ cohomology for all prime to the characteristic, assuming r ≥ dim X. The situation should be compared with that for abelian varieties A where one has Tate modules T(A) for all and these calculate H1(A, Z)for prime to the characteristic. The subject of torsion in the Chow group seems to be important from many points of view. I include in the bibliography a couple of relevant papers (Soule´ and Voisin 2005; Totaro 1997).

Lecture 6: Inﬁnitesimal methods in motivic cohomology

The infinitesimal methods developed here were used also in my work on de Rham–Witt cohomology (Bloch 1977). It is fair to say that we still do not have an adequate notion of motivic cohomology. That is, we do not have contravariant cohomology functors defined on singular schemes (e.g. on non-reduced schemes) with appropriate properties. The notion of A1-homotopy invariance which is essential in Voevodsky’s work is not what one wants. For example, if A is a non-reduced ring, then typically 1 , = × × = 1 , . HM(Spec A Z(1)) A A[t] HM(Spec A[t] Z(1)) p , K Curiously, the K-cohomology groups HZar(X q) discussed in Lecture 4 do have the correct functoriality properties, and in this lecture we examine what can be learned from the infinitesimal structure of these groups. An important step has been the work of Goodwillie (1986) computing the K-theory of nilpotent ideals in characteristic zero in terms of cyclic homology. To understand what motivic cohomology of an infinitesimal thickening might mean, the reader could consult the two rather experimental papers Bloch and

xviii Preface to the second edition

Esnault (1996, 2003). More definitive results have been obtained in Krishna and Levine (2008), Park (2007), and Rulling¨ (2007). Assuming one has a good definition of motivic cohomology, what should the “tangent space” p , = p × ε , → p , THM(X Z(q)) : Ker H (X Spec k[ ] Z(q)) H (X Z(q)) mean? (Here ε2 = 0 and the map sends ε → 0.) To begin with, one should probably not think of THM as a tangent space in the usual sense. It can be non-trivial in situations where the motivic cohomology itself is rigid, for ex- 3 , ample for HM(k Z(2)) with k a number field. Better, perhaps, to think of a non-semistable moduli functor where jumps can occur at the boundary. For example, consider the Picard scheme of (P1, {a, b}), that is isomorphism classes of line bundles on P1 with trivializations at a, b. The degree-zero part is con- stant Gm for a b, but the limit as a → b is Ga given by degree-zero line bundles on P1 with double order trivialization at a = b. One important area of open questions about these groups concerns regulators and relations with Euclidean scissors congruence groups. The regulator for usual motivic cohomology is closely related to volumes in hyperbolic space (Goncharov 1999), and it seems likely that there is a similar relation between THM and Euclidean volumes. Intuitively, this is another one of those limiting phenomena where the radius of the hyperbolic disk is allowed to go to infinity and lengths are scaled so in the limit one gets Euclidean geometry. It would be nice to have a rigorous description of how this works.

Lecture 7: Diophantine questions

At the time of these lectures, I had expected that the Chow group of zero-cycles on a rational surface would relate in some way to the zeta function of the surface. As far as I can tell, that does not happen, and I have not thought further in this direction. The reader who wants to work on diophantine questions re- garding zero-cycles and Chow groups should consult Colliot-Thel´ ene` (1995), Esnault (2003), and the references given in these papers.

Lectures 8 and 9: Regulators and values of L-functions

The whole subject of motivic cohomology, regulators, and values of L-functions remains to a large extent conjectural, but we now understand much better what should be true (Rapoport et al. 1988; Soule´ 1986; Bloch and Kato 1990). For

Preface to the second edition xix

constructions of the regulator, the reader can consult Goncharov (2005) and Bloch (1986b). Concerning the basic conjecture, I am especially attracted to the formulation given by Fontaine and Perrin-Riou (Fontaine 1992; Fontaine and Perrin-Riou 1994). To understand their idea, let X be a smooth, projective variety of dimension d over Q. (In what follows I gloss over many intractable conjectures.) Consider a motive M = hp(X)(q), where hp(X)isaChowmotive ∗ = 2d−p − + = p , as in (0.11), and write M (1) : h (X)(d q 1). Write MB HB(X Q(q)) + ⊂ for the Betti cohomology of X(C). Let MB MB be the subspace invariant = p , / 0 ∗ under the action of conjugation. Let tM : HDR(X Q(q)) F , where HDR is α + ⊗ → ⊗ de Rham cohomology. There is a natural map : MB R tM R, and (assuming certain conjectures) Fontaine and Perrin-Riou construct an exact sequence of motivic cohomology (0.15) → , ⊗ → α → 1 , ∗ ⊗ ∗ 0 HomDM(Q(0) M) R Ker (ExtDM, f (Q(0) M (1)) R) → 1 , ⊗ → α → , ∗ ⊗ ∗ → . ExtDM, f (Q(0) M) R Coker (HomDM(Q M (1)) R) 0 Here Hom and Ext are taken in the triangulated category of Voevodsky motives over Q. The subscript f refers to behavior at ﬁnite primes. As a con- sequence of (0.15) one gets a trivialization over R of the tensor product of determinant lines

(0.16) det(R HomDM, f (Q(0), M))R ∗ −1 ⊗ det(R HomDM, f (Q(0), M (1)))R ⊗ det(α) R. The various determinants have Q-structures (though α, itself, does not), so one may examine in (0.16) the ratio of the real trivialization and the rational structure. In fact, using Galois and -adic cohomology, the authors actually get a Z-structure on the left. They show that the integral conjecture in Bloch and Kato (1990) is equivalent to this ratio being given by L∗(M, 0), the ﬁrst non-vanishing term in the Taylor expansion of L(M, s) where L(M, s)isthe Hasse–Weil L-function associated to M. Well, okay, there is a lot here we do not understand, but my thought is that one might redo (0.15) working directly with the cycle complexes Zq(X, •)[−2q] and Zd+1−q(X, •)[−2(d + 1 − q)] which one should think of as concrete real- izations of motivic cohomology R ΓM(X, Z(q)) and R ΓM(X, Z(d + 1 − q)). The resulting determinant metrics could perhaps be deduced (or at least interpreted) directly from the intersection–projection map (well deﬁned in the derived category)

L Zq(X, •)[−2q] ⊗Zd+1−q(X, •)[−2(d + 1 − q)] →Z1(Spec Q, •)[−2d].

xx Preface to the second edition

Hanamura suggested this approach to understanding heights and biexten- sions. Of course, as it stands it is purely algebraic. It will be necessary to take the cone over the regulator map in some fashion. The result should be some kind of metric or related structure on the determinant of the cycle complex. This would ﬁt well with a conjecture of Soule´ (1984), which says in this context that for X regular and proper of dimension d over Spec Z,theEuler characteristic of Zp(X, •)[−2p] should be deﬁned and we should have

χ Zp , • − = − i i , = − ζ , ( (X )[ 2p]) ( 1) dimQHM(X Q(p)) ords=d−p X(s) the negative of the order of zero or pole at s = d − p of the zeta function of X.

Coda: Motives in physics

The subjects of algebraic cycles and motives have enjoyed a tremendous the- oretical development over the past 30 years. At the risk of scandalizing the reader, I would say it is high time we start looking for applications. Dirk Kreimer has been teaching me about Feynman amplitudes and pertur- bative calculations in quantum field theory. These are periods that arise, for example, in computations of scattering amplitudes. They have a strong ten- dency to be multi-zeta numbers (Bloch et al. 2006; Broadhurst and Kreimer 1997; but cf. Belkale and Brosnan 2003). The periods in question are associated to graphs. Essentially, the Kirchhoff polynomial (Bloch et al. 2006) of a graph Γ defines a hypersurface XΓ in projective space, and the Feynman am- plitude is a period of this hypersurface relative to a reference symplex. If indeed these are multi-zeta numbers it should be the case that the cohomology n , of XΓ has a big Tate piece. One knows if XΓ were smooth, then HBetti(XΓ Q) would have pure weight n, so any Tate class, i.e. any map of Hodge structures − → n , = Z( p) HBetti(XΓ Q) would necessarily be a Hodge class, i.e. n 2p.The Hodge conjecture would say that such a class comes from an algebraic cycle, 2p , i.e. a class in HM (XΓ Z(p)) via the realization map from motivic cohomology to Betti cohomology. Unfortunately (or perhaps fortunately), the XΓ are highly singular, so all one can say are that the weights of Hn are ≤ n. There are all kinds of possibilities for interesting cohomology classes coming via realization from motivic cohomology. For example, the “wheel with n spokes graph” WS(n) gives rise to a hypersurface XWS(n) of dimension 2n − 2. The primi- tive cohomology in the middle dimension for this graph is computed (Bloch et al. 2006) to be Q(−2) (independent of n). An appropriate generalization of 2n−2 , the Hodge conjecture would suggest a class in HM (XWS(n) Q(2)). The problem of computing such motivic cohomology groups can be attacked via the

Preface to the second edition xxi

combinatorics of the graph, but what has so far proved more powerful is to use classical algebro-geometric techniques to study the geometry of rank strat- iﬁcations associated to a homomorphism of vector bundles u: E → F over projective space. Ideally, knowledge of motives should provide a strong organizing force to study complex physical phenomena. Even simple motivic invariants like weight and Hodge level should help physicists understand the periods arising in their computations. More sophisticated methods involving monodromy and limiting mixed motives may give information about Landau singularities and unitarity of the S -matrix.

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